Moser iteration proceeds roughly like this: If the dimension $n$ is greater than $2$ and we assume homogeneous Dirichlet , we can proceed as follows: Using the Sobolev inequality on $\mathbb{R}^n$,
$$
c(n)\|u\|_{2n/(n-2)} \le \|\nabla u\|_2
$$
and integrating by parts, we get something roughly like this (you have to figure out what to do with all the absolute values) for each $p > 1$:
$$
\lambda\int |u|^p \ge \int |u|^{p-1}(-\Delta |u|)
= \frac{4(p-1)}{p^2}\int |\nabla |u|^{p/2}|^2
\ge \frac{4(p-1)}{p^2}c(n)\|u\|_{np/(n-2)}^p.
$$
Therefore, given $p_0 > 1$, if we set $p_k = p_0(n/(n-2))^k$, we get
$$
\|u\|_{p_{k+1}} \le \left(\frac{p_k^2}{4(p_k-1)}\frac{\lambda}{c(n)}\right)^{1/p_k}\|u\|_{p_k}.
$$
Iterating this, we get
$$
\|u\|_\infty \le C(n,p_0)\lambda^{n/(2p_0)}\|u\|_{p_0}.
$$
The power of the eigenvalue in the final estimate doesn't need to be calculated explicitly. You know what it has to be by observing that the left side is invariant under rescaling space ($\mathbb{R}^n$) and therefore the right side must be, too.